Definitions, diagrams and symbolic representation of basic geometric terms: point, line, plane, segment, distance, ray, angle, betweeness, bisectors, adjacent angles,congruent segments and angles, types of angles, etc. Basic postulates and theorems Drawing and interpreting diagrams of planar and three dimensional objects
What are the three undefined terms of geometry? What makes a good definition? How do you distinguish, symbolically and in diagrams, between lines, segments and rays? What information can and cannot be assumed from a diagram? What is the difference between congruence and equality? In a conditional, what is the hypothesis and the conclusion? What is a counterexample and when is it required?
Spatial reasoning Retaining rote information Application of rote knowledge to diagramas
Discovering Geometry- Michael Serra (DG) sections 1.1-1.3, 2.1, 2.4-2.6, 3.2-3.4 supplement intersections of points, lines, planes
Deductive reasoning: Conditional statements, Converses, inverses and contrapositives Counterexamples Properties of algebra Beginning proofs and theorems: Midpoint theorem and Angle BIsector theorem Perpendicular lines and special pairs of angles (acute, right, obtuse, straight, complementary, supplementary) Justifying statements with definitions, postulates and theorems
How are the properties of algebra (properties of equality) similar to properties of geometry (properties of congruence? How do you use the properties of algebra to justify the steps in solving an equation? How is a PROOF done and why? How do the definitions of midpoint and angle bisector differ from the statement of the theorems? How do you translate a written situation into an algebraic equation and solve?
Algebra: writing and solving equations Reasoning and logic Spatial relationships: use to diagram geoemtric figures given a verbal description
DG sections 1.5, 4.1-4.6, 4.8 DG POC investigation pgs 178-187 using GSP DG Euler Line investigation pgs 191-192 using GSP supplement writing equations of lines
Parallel lines and planes, skew lines Introduction to Geometer's Sketchpad Use of Sketchpad to discover properties of angles formed by a transversal and two or more parallel or non-parallel lines or planes Ways to show that two or more lines are parallel: show congruence of alternate interior, alternate exterior, or corresponding angles, show that same side transversal interior or exterior angles are supplementary, show coplnar lines are perpendicular to same line, show lines parallel to same line Classification of triangles by sides and angles Discover properties of angles of interior and exterior angles of a triangle, regular and non-regular polygons Patterns and sequences Coordinate geometry and algebra of lines: midpoint of a segment, slope, writing equations of lines, various forms of a linear equation, solving systems of linear equations
What are parallel lines? skew lines, perpendicular lines? When two or more parallel lines are intersected by a transversal, which lines are congruent? supplementary? What are the ways to show that two lines are parallel? How are triangles classified and what are the types of triangles? What is the sum of the exterior andinterior angles of any polygon? How are an adjacent exterior and interior angle of a polygon related? What are the three forms of a linear equation? How are slopes of parallel and perpendicular lines related? How do you write the equation of a line given: point and slope, point and parallel/perpendicular line, two points? Can you find and continue a pattern given a diagram? How do you write a rule for an arithmetic or geometric sequence given: the sequence, two consecutive or non-consecutive terms in the sequence? How do you write a sequence given the rule? How do you solve a system of equations and what does the solution mean
Algebra Spatial reasoning Knowledge of and ability to use Geometer's Sketchpad Logic and inductive/deductive reasoning
DG sections 1.4, 1.6, 5.1-5.6 Construct Quadrilaterals on GSP
Congruence of triangle and other polygons Postulate and theorems relating to triangle congruence: SAS, SSS, ASA, AAS Proving triangles congruent Using congruent triangles to prove information about angles, segments or other triangles Properties of and theorems realted to isosceles triangles Methods to prove right triangles congruent: HA, HL, LL, LA Use and Misuse of SSA (LL) property Proofs involving overalpping triangles
How is a triangle congruence proof written? How do you write the "given", "prove" and the proof AND draw a valid diagram involving triangles with only written information? What does "CPCTC" mean and when and how is it used? How is algebra used to show that triangles are/are not congruent?
Algebra Logic and deductive reasoning Spatial reasoning and representation
DG sections 11.1-11.4, 11.7 supplement properties of proportions supplement geometric mean
Medians, Altitudes, perpendicular bisectors of sides and angle bisectors in triangles Points of Concurrency in triangles ALgebra of medians, altitudes, perepndicular bisectors of sides in triangles Algebra of points of concurrency
How do altitudes, medians, angle bisectors and perpendicular bisectors of sides differ? When, if ever, are they the same line (ray)? What are concurrent points? What are the concurrent points of a triangle and what is the purpose, application, of each? What is the distance formula and how does it relate to points of concurrency? What are the incircle and circumcircle of a triangle. What are the equations of these circles? How do you algebraically find the equaiton of a median, altitude, and perpendicular bisector of a side of a triangle? How do you algebraically find the coordinates of the centroid, incenter, and circumcenter? When can you find the equation of the angles bisectors and the coordinates of the incenter?
Algebra: writing linear equations, solving systems of equations, equations of circles Geometer's Sketchpad: constructing points of concurrency,Napolean's triangle, and Euler line Spatial reasoning: using diagrams to illustrate points of concurrency
Properties of Quadrilaterals: parallelogram, rectangle, rhombus, square, trapezoid, kite Ways to show that a quadrilateral is a parallelogram, rectangle, rhombus, square, trapezoid or kite Proportional segments of a transversal created by parallel lines, proportional segments created in a triangle by a segment parallel to one side Midsegment of a triangle, median of a trapezoid Using Geometer's Sketchpad to discover properties of special quadrilaterals Coordinate proofs Algebraic investigation of median of trapezoid and midsegment of triangle
What are the properties of a parallelogram? rectangle? rhombus? square? trapezoid? isosceles trapezoid? kite? What is the quadrilateral hierarchy? How are the midsegment of a triangle and median of a trapezoid related? How do you prove (geometrically and with a coordinate proof) that a quadrialteral is a parallelogram or trapezoid? How do you prove that a parallelogram is a rectangle, rhombus or square?
Algebra Spatial reasoning Logic and deductive reasoning
Similar Polygons: Properties of Proportions, properties of similar polygons, Similar triangle postulates and theorems, proportional lengths, scale factors and similarity mappings Geometric Mean and similarity in right triangles
How do similar polygons differ from congruent polygons? What methods can be used to prove that two or more triangles are similar? How are proportions used to find missing lengths in similar polygons? What is the geometric mean between two numbers and what is its geometric interpretation?
Solving equations with proportions. Proving triangles are/are not similar. Showing that segments are proportional. Determining scale factors from graphed polygons. Finding the geometric mean. Knowing geometric mean properties in a right triangle with altitude to the hypotenuse
DG sections 1.7, 6.1-6.5, 6.7, 8.5-8.6 Supplement intersections of segments in circles
Written test Part II of Quilt Pattern Project: scale drawing
Area 1.5 week
Transformations: Mappings and functions, one-to-one mappings Isometries: translations, reflections, glide reflections, rotations Dilations: size changes and scale changes Properties of isometries Compositions of functions and mappings Rules for compositions of mappings on the coordinate plane and algebraic application Identities, inverses and symmetries.
What is a mapping? a function? one-to-one mapping?, an isometry? What compositions produce various transformations? What properties are invariant under which mappings? How do you use algebra to find the rule for a function after a particular transformation?
Using algebraic properties to write equations. Using Geometer's Sketchpad to discover properties of transformations. Applying properties of similarity to image and preimage figures. Determining number of symmetries in a particular design.
Written test. Part III of quilt pattern Project: paper discussing transformations and symmetries in pattern
Surface Area and Volume 2 weeks
Right triangles: Pythagorean theorem and converse, right triangle trig and special right triangles, applications
How do you find the length of a missing side of a right triangle? How do you show that a triangle is acute, right, or obtuse? What are the lengths of the sides of a given 30-60 right triangle or an isosceles right triangle? How do you find a missing length or angle measure in a right triangle? How do you use trig to solve application problems?
Algebra: solving equations. Simplifying radicals Interpreting application situations and writing appropriate equations, solving those equations
Circles: terminology, properties of angles and their relation to arcs(inscribed, central, vertex inside circle, vertex outside circle, vertex on circle), relationship between lengths of segments (secants, chords, tangents),Area and circumference
What are the various parts of a circle, the types of arcs, the different segments and lines that relate to a circle? What is the relationship between chords, secants and tangents AND their intercepted arcs? What is the relationship between different segments and partial segments in a circle? How do you find the circumference and area of a circle?
Algebra: setting up and solving equations Recognition and spatial relationship skills: identifying parts of a circle and relationship between parts, identifying arcs, intercepted angles, tangents, chords, secants Using right triangle trig to solve problems
Table of Contents
3 weeks
Basic postulates and theorems
Drawing and interpreting diagrams of planar and three dimensional objects
What makes a good definition?
How do you distinguish, symbolically and in diagrams, between lines, segments and rays?
What information can and cannot be assumed from a diagram?
What is the difference between congruence and equality?
In a conditional, what is the hypothesis and the conclusion? What is a counterexample and when is it required?
Retaining rote information
Application of rote knowledge to diagramas
supplement intersections of points, lines, planes
4.5 weeks
Counterexamples
Properties of algebra
Beginning proofs and theorems: Midpoint theorem and Angle BIsector theorem
Perpendicular lines and special pairs of angles (acute, right, obtuse, straight, complementary, supplementary)
Justifying statements with definitions, postulates and theorems
How do you use the properties of algebra to justify the steps in solving an equation?
How is a PROOF done and why?
How do the definitions of midpoint and angle bisector differ from the statement of the theorems?
How do you translate a written situation into an algebraic equation and solve?
Reasoning and logic
Spatial relationships: use to diagram geoemtric figures given a verbal description
DG POC investigation pgs 178-187 using GSP
DG Euler Line investigation pgs 191-192 using GSP
supplement writing equations of lines
2.5 weeks
Introduction to Geometer's Sketchpad
Use of Sketchpad to discover properties of angles formed by a transversal and two or more parallel or non-parallel lines or planes
Ways to show that two or more lines are parallel: show congruence of alternate interior, alternate exterior, or corresponding angles, show that same side transversal interior or exterior angles are supplementary, show coplnar lines are perpendicular to same line, show lines parallel to same line
Classification of triangles by sides and angles
Discover properties of angles of interior and exterior angles of a triangle, regular and non-regular polygons
Patterns and sequences
Coordinate geometry and algebra of lines: midpoint of a segment, slope, writing equations of lines, various forms of a linear equation, solving systems of linear equations
When two or more parallel lines are intersected by a transversal, which lines are congruent? supplementary?
What are the ways to show that two lines are parallel?
How are triangles classified and what are the types of triangles?
What is the sum of the exterior andinterior angles of any polygon? How are an adjacent exterior and interior angle of a polygon related?
What are the three forms of a linear equation?
How are slopes of parallel and perpendicular lines related?
How do you write the equation of a line given: point and slope, point and parallel/perpendicular line, two points?
Can you find and continue a pattern given a diagram?
How do you write a rule for an arithmetic or geometric sequence given: the sequence, two consecutive or non-consecutive terms in the sequence?
How do you write a sequence given the rule?
How do you solve a system of equations and what does the solution mean
Spatial reasoning
Knowledge of and ability to use Geometer's Sketchpad
Logic and inductive/deductive reasoning
Construct Quadrilaterals on GSP
2-2.5 weeks
Postulate and theorems relating to triangle congruence: SAS, SSS, ASA, AAS
Proving triangles congruent
Using congruent triangles to prove information about angles, segments or other triangles
Properties of and theorems realted to isosceles triangles
Methods to prove right triangles congruent: HA, HL, LL, LA
Use and Misuse of SSA (LL) property
Proofs involving overalpping triangles
How do you write the "given", "prove" and the proof AND draw a valid diagram involving triangles with only written information?
What does "CPCTC" mean and when and how is it used?
How is algebra used to show that triangles are/are not congruent?
Logic and deductive reasoning
Spatial reasoning and representation
supplement properties of proportions
supplement geometric mean
3 weeks
End 1st Semester
Points of Concurrency in triangles
ALgebra of medians, altitudes, perepndicular bisectors of sides in triangles
Algebra of points of concurrency
What are concurrent points? What are the concurrent points of a triangle and what is the purpose, application, of each?
What is the distance formula and how does it relate to points of concurrency?
What are the incircle and circumcircle of a triangle. What are the equations of these circles?
How do you algebraically find the equaiton of a median, altitude, and perpendicular bisector of a side of a triangle? How do you algebraically find the coordinates of the centroid, incenter, and circumcenter?
When can you find the equation of the angles bisectors and the coordinates of the incenter?
Geometer's Sketchpad: constructing points of concurrency,Napolean's triangle, and Euler line
Spatial reasoning: using diagrams to illustrate points of concurrency
DG GSP tesselation activity pgs 408-409
Project
4-4.5 weeks
Ways to show that a quadrilateral is a parallelogram, rectangle, rhombus, square, trapezoid or kite
Proportional segments of a transversal created by parallel lines, proportional segments created in a triangle by a segment parallel to one side
Midsegment of a triangle, median of a trapezoid
Using Geometer's Sketchpad to discover properties of special quadrilaterals
Coordinate proofs
Algebraic investigation of median of trapezoid and midsegment of triangle
What is the quadrilateral hierarchy?
How are the midsegment of a triangle and median of a trapezoid related?
How do you prove (geometrically and with a coordinate proof) that a quadrialteral is a parallelogram or trapezoid? How do you prove that a parallelogram is a rectangle, rhombus or square?
Spatial reasoning
Logic and deductive reasoning
Supplement coordinate geometry
supplement trig applications
4.5 weeks
Properties of Proportions, properties of similar polygons, Similar triangle postulates and theorems, proportional lengths, scale factors and similarity mappings
Geometric Mean and similarity in right triangles
What methods can be used to prove that two or more triangles are similar?
How are proportions used to find missing lengths in similar polygons?
What is the geometric mean between two numbers and what is its geometric interpretation?
Proving triangles are/are not similar.
Showing that segments are proportional.
Determining scale factors from graphed polygons.
Finding the geometric mean. Knowing geometric mean properties in a right triangle with altitude to the hypotenuse
Supplement intersections of segments in circles
Part II of Quilt Pattern Project: scale drawing
1.5 week
Mappings and functions, one-to-one mappings
Isometries: translations, reflections, glide reflections, rotations
Dilations: size changes and scale changes
Properties of isometries
Compositions of functions and mappings
Rules for compositions of mappings on the coordinate plane and algebraic application
Identities, inverses and symmetries.
What compositions produce various transformations?
What properties are invariant under which mappings?
How do you use algebra to find the rule for a function after a particular transformation?
Using Geometer's Sketchpad to discover properties of transformations. Applying properties of similarity to image and preimage figures. Determining number of symmetries in a particular design.
Part III of quilt pattern Project: paper discussing transformations and symmetries in pattern
2 weeks
What are the lengths of the sides of a given 30-60 right triangle or an isosceles right triangle?
How do you find a missing length or angle measure in a right triangle? How do you use trig to solve application problems?
Simplifying radicals
Interpreting application situations and writing appropriate equations, solving those equations
3 weeks
Recognition and spatial relationship skills: identifying parts of a circle and relationship between parts, identifying arcs, intercepted angles, tangents, chords, secants
Using right triangle trig to solve problems